The exterior angles of a triangle is `130^(@)` and the two interior opposite angle are in the ratio 6:7. find the angles of the triangle.

Answer

To solve the problem, we need to find the angles of a triangle given that one of its exterior angles is \(130^\circ\) and the two interior opposite angles are in the ratio \(6:7\).

### Step-by-Step Solution:

1. **Understanding the Exterior Angle**:
The exterior angle of a triangle is equal to the sum of the two opposite interior angles. In this case, the exterior angle \( \angle CAD \) is given as \(130^\circ\).

2. **Finding the Interior Angle**:
According to the property of triangles, the sum of the interior angle \( \angle BAC \) and the exterior angle \( \angle CAD \) is \(180^\circ\).
\[ \angle BAC + \angle CAD = 180^\circ \]
Substituting the value of the exterior angle:
\[ \angle BAC + 130^\circ = 180^\circ \]
Therefore,
\[ \angle BAC = 180^\circ - 130^\circ = 50^\circ \]

3. **Setting Up the Ratio**:
Let the two interior angles \( \angle ABC \) and \( \angle ACB \) be represented as \(6x\) and \(7x\) respectively, based on the ratio \(6:7\).

4. **Applying the Angle Sum Property**:
The sum of the angles in a triangle is \(180^\circ\). Therefore, we can write:
\[ \angle BAC + \angle ABC + \angle ACB = 180^\circ \]
Substituting the known values:
\[ 50^\circ + 6x + 7x = 180^\circ \]
Simplifying this gives:
\[ 50^\circ + 13x = 180^\circ \]

5. **Solving for \(x\)**:
Rearranging the equation:
\[ 13x = 180^\circ - 50^\circ \]
\[ 13x = 130^\circ \]
Dividing both sides by 13:
\[ x = \frac{130^\circ}{13} = 10^\circ \]

6. **Finding the Angles**:
Now substituting \(x\) back to find the angles:
- \( \angle ABC = 6x = 6 \times 10^\circ = 60^\circ \)
- \( \angle ACB = 7x = 7 \times 10^\circ = 70^\circ \)

7. **Final Angles of the Triangle**:
The angles of the triangle are:
- \( \angle BAC = 50^\circ \)
- \( \angle ABC = 60^\circ \)
- \( \angle ACB = 70^\circ \)

### Summary of the Angles:
- \( \angle BAC = 50^\circ \)
- \( \angle ABC = 60^\circ \)
- \( \angle ACB = 70^\circ \)